SOLUTION: x-3y=7 2x-5y=13 solve by substitution

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Question 150378: x-3y=7
2x-5y=13
solve by substitution
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!


Start with the given system of equations:

system%28x-3y=7%2C2x-5y=13%29



Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.




So let's isolate y in the first equation

x-3y=7 Start with the first equation


-3y=7-x Subtract x from both sides


-3y=-x%2B7 Rearrange the equation


y=%28-x%2B7%29%2F%28-3%29 Divide both sides by -3


y=%28%28-1%29%2F%28-3%29%29x%2B%287%29%2F%28-3%29 Break up the fraction


y=%281%2F3%29x-7%2F3 Reduce



---------------------

Since y=%281%2F3%29x-7%2F3, we can now replace each y in the second equation with %281%2F3%29x-7%2F3 to solve for x



2x-5highlight%28%28%281%2F3%29x-7%2F3%29%29=13 Plug in y=%281%2F3%29x-7%2F3 into the first equation. In other words, replace each y with %281%2F3%29x-7%2F3. Notice we've eliminated the y variables. So we now have a simple equation with one unknown.



2x%2B%28-5%29%281%2F3%29x%2B%28-5%29%28-7%2F3%29=13 Distribute -5 to %281%2F3%29x-7%2F3


2x-%285%2F3%29x%2B35%2F3=13 Multiply


%283%29%282x-%285%2F3%29x%2B35%2F3%29=%283%29%2813%29 Multiply both sides by the LCM of 3. This will eliminate the fractions (note: if you need help with finding the LCM, check out this solver)



6x-5x%2B35=39 Distribute and multiply the LCM to each side



x%2B35=39 Combine like terms on the left side


x=39-35Subtract 35 from both sides


x=4 Combine like terms on the right side





-----------------First Answer------------------------------


So the first part of our answer is: x=4









Since we know that x=4 we can plug it into the equation y=%281%2F3%29x-7%2F3 (remember we previously solved for y in the first equation).



y=%281%2F3%29x-7%2F3 Start with the equation where y was previously isolated.


y=%281%2F3%29%284%29-7%2F3 Plug in x=4


y=4%2F3-7%2F3 Multiply


y=-1 Combine like terms and reduce. (note: if you need help with fractions, check out this solver)



-----------------Second Answer------------------------------


So the second part of our answer is: y=-1









-----------------Summary------------------------------

So our answers are:

x=4 and y=-1

which form the point








Now let's graph the two equations (if you need help with graphing, check out this solver)


From the graph, we can see that the two equations intersect at . This visually verifies our answer.




graph of x-3y=7 (red) and 2x-5y=13 (green) and the intersection of the lines (blue circle).